In most commercial and industrial activities in fields such as manufacturing, retail, financial markets, call centers, energy systems, forecasting is one of the most important operational concerns. Forecasts are fundamental inputs to decision making and planning.
Various forecasting methods use the time series history of a variable for which forecasts of future values are desired. A time series is a set of observations of a variable that are ordered and identified by the time of observation (e.g. call volumes arriving at a call center recorded every half hour with the time of the day and date, average service time for the calls answered in a time interval, energy consumption every half hour with the time of the day and date).
Time series methods use the past values of a time series to identify patterns in the data to generate forecasts. Patterns that may exist in a time series include trend, seasonal patterns, and cyclical pattern. Most time series methods focus on modeling trend and only one seasonal cycle. For example, the exponential smoothing technique has three versions: i) single exponential smoothing generating smoothed values without any trend or seasonal pattern, ii) double exponential smoothing generating forecasts with smoothed values adjusted for trend, and iii) triple exponential smoothing generating forecasts by adjusting smoothed values for trend and one seasonal pattern (Makridakis, S., et al., 1998).
Although there are various forecasting methods that can model trend and seasonal pattern, almost all of the forecasting methods are limited to at most one seasonal pattern. These methods can model one seasonal pattern, but not any two or three at the same time. If a seasonal pattern is not modeled and included in forecasting, however, the accuracy of the resulting forecasts will be lower than otherwise. In call centers, for example, incoming calls may vary not only during the course of a day (intra-day variation) but also over the days of a week, and over different weeks of a year. In these centers, forecasting methods not incorporating multiple seasonal patterns will ignore one or more of the intraday, the day-of-week and the week-of-year patterns, and will result in forecasts with poor accuracy. Other examples of time series with multiple seasonal patterns include electrical energy consumption, stock prices, and activities in banks, hospitals, police stations, and other service, production, and financial systems.
The Autoregressive Integrated Moving Average (ARIMA) procedure proposed by Box and Jenkins (Box, et al., 1994), is known to have the capability to model multiple seasonal patterns. Another forecasting approach for time series with double seasonal patterns recently developed by Taylor (2003). It involves an extension of the triple exponential smoothing method (also know as Holt-Winters' method). The approach is based on the multiplicative version of the method that includes the seasonal effects in the form of multipliers applied to the trend and level based forecasts. The additive model, on the other hand, includes the seasonal effect in the form of an additive factor applied to the trend and level based forecasts. The additive seasonal effect modeling is more appropriate if the seasonal effect does not change with the current mean of a time series.
Given the limited availability of the forecasting methods that can be used for time series with multiple seasonal patterns, an objective of the present invention is to provide a new method for this type of time series using the additive version of the exponential smoothing method. Another objective of the present invention is to provide two general methods for time series and forecast processing that can be used with any forecasting method to capture multiple seasonal patterns. The present invention further comprises a computer system for applying, on one or more computers, the methods of the present invention to time series exhibiting multiple seasonal patterns.